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Q-I-J-are-two-ideals-of-commutative-ring-R-prove-that-I-J-I-J-m-n-note-I-x-R-n-N-x-n-I-




Question Number 174684 by mnjuly1970 last updated on 08/Aug/22
  Q:  I , J  are two ideals of  commutative       ring , ( R ,⊕,   ) .prove that :                (√( I ∩ J ))  =^?  (√( I ))  ∩  (√( J ))     m.n      note : (√(I )) = { x ∈ R ∣ ∃ n∈ N , x^( n)  ∈ I }
Q:I,Jaretwoidealsofcommutativering,(R,,).provethat:IJ=?IJm.nnote:I={xRnN,xnI}
Answered by mindispower last updated on 09/Aug/22
I∩J ideal  (√(I∩J))={x∈R ∣∃n∈N x^n ∈I∩J}  x^n ∈I⇒x∈(√I)  x^n ∈J⇒x∈(√j)  ⇒x∈(√I)∩(√J)⇒(√(I∩J))⊂(√I)∩(√J)  x∈(√I)∩(√J)⇒∃n,m ∣x^n ∈I ,x^m ∈J  x^(n+m) ∈I“ x^n .x^m ∈x^m .I=I,“x^(n+m) ∈J,x^(m+n) =x^n .x^m ∈x^n J=J}  ⇒x^(n+m) ∈I∩J⇒x∈(√(I∩J))⇒(√I)∩(√J)⊂(√(I∩J))  (√(I∩J))=(√I)∩(√J)
IJidealIJ={xRnNxnIJ}xnIxIxnJxjxIJIJIJxIJn,mxnI,xmJxn+mIxn.xmxm.I=I,xn+mJ,xm+n=xn.xmxnJ=J}xn+mIJxIJIJIJIJ=IJ
Commented by mnjuly1970 last updated on 14/Aug/22
  thanks alot  sir power...
thanksalotsirpower

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